research: all-endpoint nonlinear recovery obstruction for PR2918

research/rank7-terminal-information-barrier/README.md

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+# A nonlinear recovery obstruction at every rank-seven endpoint
+
+For the two PR2894 providers pinned by PR2918, **no endpoint of rank at most
+seven permits exact recovery of any nonconstant function of the eligible
+entrance state**. This remains true with arbitrary nonlinear decoding,
+arbitrary recoding confined to eligible entrances, and arbitrary appended
+processing whose complete data dependencies obey monotone frame inclusion.
+All inputs retain the independent arbitrary-dirty contract. The word, entrance
+cut, protected inputs and observation interface are fixed.
+
+The entrance cut is immediately after the 84-event prelude. The separate
+target-only center cut uses the inherited, checked `center_frames` convention;
+this package does not reconstruct extra physical moves for its unspecified
+frames. Output registers and frames in the suffix are fixed independently of
+data, with all branch and control dependencies included.
+
+This strengthens PR2918's one-helper, one-endpoint, scalar paired-boundary
+obstruction. It concerns deterministic recovery for every input, not statistical
+mutual information. [PROOF.md](PROOF.md) gives the complete theorem and limits.
+
+## Exact evidence
+
+Each provider has 520 live non-target registers and 226 eligible entrances.
+We enumerate all **43,435** rank-seven subspaces of `GF(2)^9`, obtaining **239**
+distinct complete terminal supports, of size at most 16. Every support has a
+square purely scalar minor on protected input columns with determinant a unit
+of `Z[1/2]`. The census covers all 58 eligible helper slots whose current terminal
+frames have rank at most seven. Lower-rank endpoints follow by containment.
+
+| Determinant | Support classes, per provider |
+| --- | ---: |
+| 1 | 157 |
+| -1 | 34 |
+| 1/2 | 11 |
+| -1/2 | 17 |
+| 1/4 | 8 |
+| -1/4 | 12 |
+
+The empty support has determinant one by convention. Each nonempty certificate
+includes the entire terminal support, a representative annihilator, its endpoint
+multiplicity, and the forbidden input columns. All nonempty operator coefficients
+on the chosen blocks are zero. Thus these certificates remain valid under every
+invertible linear specialization of the formal warps on every dyadic module.
+
+The mathematical reason is simple. In `y=A h+B z`, protected inputs `z` can
+produce every observation because `B` is invertible. Any eligible-input change
+can be canceled by `z -> z+B^-1 A(h-h')`. No decoder can distinguish those
+states. Monotone frame dependencies confine any output at endpoint `E` to the
+already masked terminal support `R(E)`.
+
+The known PR2908 rank-eight paired extraction passes both verification paths.
+For every nonzero dyadic module, eight is the sharp endpoint threshold in this
+frozen interface
+when the permitted eligible-input preprocessing is included. This does not
+assert that an output-only suffix can attain the threshold.
+
+## Reproduction and verification scope
+
+```sh
+python3.12 scripts/research_package.py --package research/rank7-terminal-information-barrier
+```
+
+The two direct commands are:
+
+```sh
+python3.12 -I -O -B research/rank7-terminal-information-barrier/verify.py
+python3.12 -I -O -B research/rank7-terminal-information-barrier/verify_independent.py
+```
+
+`verify.py` authenticates and directly executes PR2918's existing forward
+operator/frame engine, then regenerates the census and checks the fixed
+`certificates.json`. It checks both inverse products for every minor, eight
+rejecting controls per provider, a row/column reordering control, and the
+rank-eight construction. This path deliberately reuses the predecessor engine.
+
+`verify_independent.py` imports no predecessor code. It parses the two source
+providers, propagates 164 terminal covectors backward, enumerates endpoints by
+two-row reduced echelon normal matrices, and verifies complete scalar minors
+using fraction-free Bareiss determinants. It checks seven rejecting controls per
+provider and separately reconstructs the rank-eight positive pairing by a row
+pullback and inverse column. Its adjoint approach follows the credited PR2918
+checker; the implementation and all-endpoint coverage are separate.
+
+Both paths use exact rational arithmetic and explicit checks that remain active
+under Python `-O`. This is finite exhaustive computational evidence plus a
+written all-parameter argument. It is not a formal-kernel proof or independent
+human review. Four authenticated predecessor files are live inputs, declared in
+`VERIFY.json` and pinned by `SOURCES.json`; they are not duplicated here.
+
+## Prior work and importance
+
+PR2918 is pinned at `9360ba7312cf8961e672e9eb26c612719028d39d`; its sources bind
+PR2908 at `51309592971e3b5be935599eed2a95404dc7781e` and PR2894 at
+`18d03c447886895ef9abf888398178e0428533ca`.
+[PR2979](https://github.com/CrocSwap/integer-mult-bounds/pull/2979), head
+`be191346391661f4d35fc0d1508d0a6029572576`, already uses the all-43,435-endpoint
+pairing screen on later words and reports a rank-seven positive example for
+PR2917's changed word. We do not claim the enumeration technique or an
+all-endpoint linear screen is new. The scoped strengthening here is the complete
+unit-minor census on the old parent and its **nonlinear causal recovery
+obstruction**, together with explicit side-information and repair requirements.
+The underlying quotient and indistinguishability principles are standard.
+No comprehensive literature novelty claim is made.
+
+This excludes an entire family of endpoint/postprocessing repairs of the frozen
+construction. It gives a necessary boundary for further design; it supplies no
+new multiplication algorithm, improved kappa, paid recovery, owner allocation,
+or native/outer theorem admission. A different middle word, additional retained
+information, different entrance dependencies, a new warp reset, or a changed
+observation contract can escape the theorem. Pooling observations from
+incomparable low-rank endpoints is outside the single-common-endpoint result.
+
+## Open next questions
+
+The structured questions are in `TASK.json`. The useful next step is to identify
+one paid change that destroys a certified nuisance block and then price the
+entire resulting construction. A matrix-rank repair bound alone neither realizes
+such a change nor proves it improves the complete recursive cost. Existing
+changed-word rank-seven examples should be used as positive controls before
+starting any further synthesis.
+
+## Credit
+
+Prepared for Ross Richardson with substantial OpenAI Codex assistance, including
+separately assigned helper-boundaries, observation-kernels and exact-verification
+specialists. PR2918 is Ross Richardson's scoped obstruction; Jeremy Diviney's
+PR2908 supplies the rank-eight extraction, and Jeremy Kothe's PR2894 supplies the
+parent words. Marius-Paul Dumitrean's PR2979 supplies the prior all-endpoint
+screening comparison.
+Earlier contributions and license notices remain in
+`research/helper408-rank7-pairing/NOTICE.md` and its `sources/CREDITS.md`.
+This package is under the repository's Apache-2.0 license. No new copies of
+predecessor source files are vendored.

research/rank7-terminal-information-barrier/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "997e6a9ca7b4f95d50e366f61412fd640d72a88a",
+  "files": [
+    {
+      "path": "research/helper408-rank7-pairing/verify_boundary.py",
+      "usage": "input",
+      "git_blob": "39756c56dd647c665e293506899db7e4b6ddb59c",
+      "sha256": "56ca12652c45f57c74341a06be2dbce4617415656e65dfe08223a4afcf8de541"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/sources/data/PARENT-base.json.gz",
+      "usage": "input",
+      "git_blob": "0ca814951069c6f029eb375f3ed94053ba45bc63",
+      "sha256": "c20a470aec0f3e61833cb41fe190639eb358f58017143d2336c55c80fba265eb"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/sources/data/PARENT-extra.json.gz",
+      "usage": "input",
+      "git_blob": "5c9e40118c599163571c10003db42eefa6f042a9",
+      "sha256": "db34e39876241087b581ed9374c53e9cd393e3e86ef6103bfb54f3077c3d2761"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/sources/data/TRANSFORM.json",
+      "usage": "input",
+      "git_blob": "0e81cdc3e328fa8eedbdadda8f5065a3e3dc3dd6",
+      "sha256": "1d38f0369fce0b892d202ce2e5fd7ea95d9a95d3f80734b672cddd5cb93ce8ca"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/BOUNDARY.md",
+      "usage": "provenance",
+      "git_blob": "fd7de6b25bf3c642b37d060716dd44f7be100bda",
+      "sha256": "a59846249563d7229d471eab3b127f7333ff43fa5abe74b2555fc718c3a412f6"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/INDEPENDENT.md",
+      "usage": "provenance",
+      "git_blob": "ecf922c35a00234b4bd8353bd8870691bea12c0e",
+      "sha256": "23981dd600ada4f9716800c51e8c0e397c86c557ace16cc6dcce5ab3e51fd228"
+    },
+    {
+      "path": "research/helper408-rank7-pairing/NOTICE.md",
+      "usage": "provenance",
+      "git_blob": "ec74fb826b1d5e33690d40848e23ecc5ec505146",
+      "sha256": "f37dd53e7ebb6af7a27084fff7f7b9e4112d020d207177e442a6182fcbdd197a"
+    },
+    {
+      "path": "research/rank7-joint-endframes/README.md",
+      "usage": "provenance",
+      "git_blob": "ad4fde698389d02543df632d46b54dcd93e53a97",
+      "sha256": "d00d123fc6cb7da627a8db1b8ae95b732ef7c55f802a784059de982159548634"
+    }
+  ]
+}

research/rank7-terminal-information-barrier/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 6,
+  "base_commit": "997e6a9ca7b4f95d50e366f61412fd640d72a88a",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Generalize PR2918 to all endpoints of rank at most seven on its two pinned providers, including arbitrary nonlinear decoding and monotone causal suffixes, with independent exact unit-minor certificates.",
+  "scope_change_reason": "Explicit user-assigned generalisation and publication review of PR2918. This keeps the old word and varies terminal readout/postprocessing, which is a scope-change trigger in the original card. The submission establishes a scoped obstruction; it does not satisfy the card's positive changed-interface construction or stronger-budget acceptance gate.",
+  "bottleneck_link": "The fixed provider cannot yield any new eligible-input function through any rank-seven terminal readout, so endpoint-only repairs cannot change its helper inventory. Any candidate must alter a certified premise and pay for that change.",
+  "budget_minutes": 240,
+  "stop_rule": "Finish the exact pinned census, independent verification and written causal proof; do not expand into changed-word synthesis or claim a new multiplication bound.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md, certificates.json, primary forward replay and independently written reverse-adjoint/RREF/Bareiss verification with rejecting controls and a rank-eight positive calibration.",
+  "author": "Ross Richardson",
+  "ai_assistance": "Substantial OpenAI Codex assistance, including separately assigned mathematical specialists. No independent human or formal-kernel review claimed.",
+  "open_questions": [
+    {
+      "question": "Which paid change to the fixed word or retained-source interface can destroy the certified nuisance saturation at a rank-seven endpoint?",
+      "why_it_matters": "Every endpoint-only monotone postprocessing repair is excluded, including nonlinear decoding; an escape must change a premise.",
+      "first_step": "Select one known changed-word rank-seven positive example, compute the induced change to the relevant nuisance block, and bind every changed entrance, source, restoration and recursive cost before proposing a new construction.",
+      "blocked_by": "This package proves a necessary obstruction, not a paid construction or a complete cost improvement."
+    }
+  ],
+  "scope_key": "pr2918-all-rank7-unit-nuisance-nonlinear-monotone-barrier",
+  "no_new_kappa": true,
+  "predecessors": [
+    {
+      "pr": 2918,
+      "head": "9360ba7312cf8961e672e9eb26c612719028d39d"
+    },
+    {
+      "pr": 2979,
+      "head": "be191346391661f4d35fc0d1508d0a6029572576"
+    }
+  ]
+}

research/rank7-terminal-information-barrier/VALIDATION.md

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+# Local validation receipt
+
+Validated on 2026-10-17 UTC using Python 3.12, on base
+`997e6a9ca7b4f95d50e366f61412fd640d72a88a`:
+
+```sh
+python3.12 scripts/research_package.py --package research/rank7-terminal-information-barrier
+```
+
+Result: `PASS_REGISTERED_RESEARCH_PACKAGE`, outcome `negative-result`.
+Both registered commands ran with `-I -O -B` and returned success:
+
+- `PASS_ALL_RANK_LE7_ENDPOINTS_DYADIC_NUISANCE_SURJECTIVITY`
+- `PASS_INDEPENDENT_TERMINAL_INFORMATION_BARRIER`
+
+Both authenticated providers produced 43,435 endpoints, 239 support classes,
+maximum support size 16, 58 eligible low-rank slots, and the determinant counts
+in the README. The primary path checked both inverse products for each minor,
+eight rejecting controls per provider and the actual rank-eight pre/post maps.
+The independent path checked seven rejecting controls with exact expected
+diagnostics, complete scalar blocks over the dyadic ring, and a separate
+rank-eight inverse-column/adjoint-row pairing. The fixed certificate SHA256 is
+`aa720fe2591cc511b58a0faea7752aba3c72ef90164b606eb9c0e437b9fa8863`.
+
+A separate specialist review compared the two implementations on all 4,583
+operations, 520 frame spans, 165 complete output rows (164 of rank at most seven
+plus register42), 239 support multiplicities, and the rank-eight inverse vector.
+It found exact agreement. Mathematical review confirmed the causal and
+indistinguishability arguments and led to the explicit linear-encoding
+restriction on the secondary dimension bounds.
+
+A fresh pre-submission review additionally compared all 520 complete operator
+rows and 11,912 coefficients on each provider, all 520 frames, the endpoint
+partition and the rank-eight inverse vector. Fourteen disposable variants with
+named validation checks removed were detected by the existing controls; a valid
+alternative gzip encoding separately tested source-byte authentication, and a
+noncommuting two-generator fixture checked word direction and cancellation.
+This was additional author-review work, not a third registered independent
+verifier; its scripts and receipts are retained by the author outside the package.
+
+The pre-submission mathematical fixes make the post-prelude entrance cut and
+inherited center-cut frame convention explicit, fix the dependency graph and
+output identities against control side channels, require joint ancillary
+independence, and qualify sharpness by a nonzero module. The task card now
+declares changed scope: this negative result does not satisfy the original
+positive-construction acceptance gate. Fresh mathematical and packaging reviews
+found no remaining issue within those stated limits.
+
+This receipt records local computation and AI-assisted mathematical review.
+It is not hosted CI, human peer review, formal-kernel verification, a full native
+supplier replay, or admission of a new multiplication bound. The original selected
+certificate and all predecessor sources are unchanged.

research/rank7-terminal-information-barrier/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/helper408-rank7-pairing/verify_boundary.py",
+    "research/helper408-rank7-pairing/sources/data/PARENT-base.json.gz",
+    "research/helper408-rank7-pairing/sources/data/PARENT-extra.json.gz",
+    "research/helper408-rank7-pairing/sources/data/TRANSFORM.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Pinned forward replay; complete endpoint census; dyadic-unit scalar minors and inverse products; eight rejecting controls per provider; rank-eight calibration."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify_independent.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Separate source parser and reverse adjoint; RREF endpoint coverage; Bareiss determinants; seven rejecting controls per provider; independent rank-eight pairing."
+    }
+  ]
+}